Earlier this year, I was
introduced to Paul Lockhart’s remarkable text, A Mathematician’s Lament: How School Cheats Us Out of Our
Most Fascinating and Imaginative Art Form. Lockhart persuasively
advocates for mathematics teaching and a mathematics curriculum that provides
students with opportunities to ask their own questions and devise their own
methods for answering these questions. Taking Lockhart’s message to heart, I
asked my eighth-grade daughter, Cassady, to construct a mathematics task that
fosters creativity among students. My conversation with Cassady surprised me. Both
she and Lockhart have encouraged me to reconsider the nature of mathematics and
the way that we share it with students in the middle grades.
I began by asking
Cassady whether mathematics was creative. An edited version of the hourlong
conversation is highlighted in the following 10-minute video (http://bit.ly/mathcreative).
Cassady remarked that mathematics
in the middle grades is “like a formula, and there’s one correct answer.” In
response, she provided me with the following task:
John had 10
sticks. They were 12 cm, 2 cm, 4 cm, 9 cm, 20 cm, 6 cm, 15 cm, 5 cm, 10 cm, and
1 cm long. Can John make a polygon using these 10 sticks? If so, draw it. If
not, explain why not.
Remarkably, the task was not
familiar to Cassady. It had not been posed to her in a previous class. The task
was truly her own. In fact, she didn’t know if such a polygon could be built. When
I asked Cassady if her classmates had opportunities to pose and solve their own
problems, Cassady remarked, “We have kind of outgrown it. We don’t do story
problems anymore . . . we haven’t done that since third grade.”
During the next 30 minutes, Cassady
and I worked collaboratively on her problem. Unfortunately, we had no hands-on
manipulatives, so we attacked the problem using GeoGebra (http://www.geogebra.org/cms/en/), a
free dynamic mathematics software program. A version of her sketch can be
accessed on GeoGebraTube at http://www.geogebratube.org/student/m112159.
Cassady began by plotting 8 of the
10 vertices as lattice points on a square grid. She arranged consecutive
vertices to form right angles. This made it easier for her to satisfy the
length requirements of the task. I’ve labeled these points consecutively as A through I (see fig. 1).
Fig. 1 We approached the
solution to this problem using GeoGebra.
Plot of first 8 vertices
(b) Constructing a circle to plot 2 remaining vertices
With 2 lengths remaining (namely,
10 cm and 15 cm), Cassady constructed a circle with radius 10 units centered at
the 8th vertex (point I). Point J was constructed on the circle. This
step ensured that segment IJ, the 9th
side of the polygon, had length 10 units. Next, she constructed JA as the 10th side, dragging J until JA had the desired length of 15 units.
strategy also highlighted Cassady’s strategic use of technology (one of the
Common Core’s Standards for Mathematical Practice is Standard 5: Use appropriate
tools strategically). GeoGebra played an invaluable role in the problem-solving
process. Using a compass and ruler to construct the location of point J would have been time-consuming and
error-laden without the software. However, GeoGebra did not trivialize the
task. Rather, the problem required a good deal of creative thinking with the software. The idea to use a
circle was a “creative breakthrough” that involved considerable mathematics
know-how and outside-the-box thinking. In the journey to the solution, Cassady
and I discussed a number of interesting concepts, including the triangle
inequality and slopes of parallel lines.
Math Is Creative
At the conclusion
of our conversation, Cassady noted that our mathematics work was “definitely”
more creative than the activities she does in school. She noted that she
“really likes problems where she doesn’t know if there’s a definite answer.” The
experience has encouraged me to ask more questions of my students regarding
their own learning needs, listening carefully to their answers, then acting on
them. As Lockhart notes, “The only people who understand what is going on are
the ones most often blamed and least often heard: the students” (p. 11).
Todd Edwards is an associate professor of mathematics education at Miami
University in Oxford, Ohio. He is the coeditor of Contemporary Issues in Technology and Mathematics Teacher Education,
executive editor of the North American
GeoGebra Journal, and codirector of the GeoGebra Institute of Ohio. His
research interests focus on the teaching and learning of school mathematics
with technology (specifically, dynamic mathematics software), ethical issues
surrounding the use of free software and the free software movement, and
writing as a vehicle to learn mathematics at all levels of instruction.